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Portions not contributed by visitors are Copyright 2018 Tangient LLC
TES: The largest network of teachers in the world
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5L 2s - "diatonic"
One way of distinguishing the "diatonic" scale is by considering it a moment of symmetry scale produced by a chain of "fifths". This will include 12edo's diatonic scale along with the Pythagorean diatonic scale and meantone systems, while excluding just intonation scales that use more than one size of "tone".
It may be misleading to call 5L 2s "diatonic," since other scales called diatonic can be arrived at different ways (through just intonation procedures for instance, or with tetrachords). Also, a composer working with a 5L 2s scale may choose to do something very different than typical diatonic music.
substituting step sizes
The 5L 2s MOS scale has this generalized form.
L L s L L L s
Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice.
2 2 1 2 2 2 1
When L=3, s=1, you have 17edo:
3 3 1 3 3 3 1
When L=3, s=2, you have 19edo:
3 3 2 3 3 3 2
When L=4, s=1, you have 22edo:
4 4 1 4 4 4 1
When L=4, s=3, you have 26edo:
4 4 3 4 4 4 3
When L=5, s=1, you have 27edo:
5 5 1 5 5 5 1
When L=5, s=2, you have 29edo:
5 5 2 5 5 5 2
When L=5, s=3, you have 31edo:
5 5 3 5 5 5 3
When L=5, s=4, you have 33edo:
5 5 4 5 5 5 4
So you have scales where L and s are nearly equal, which approach 7edo:
1 1 1 1 1 1 1
And you have scales where s becomes so small it approaches zero, which would give us 5edo:
1 1 0 1 1 1 0 or 1 1 1 1 1
a continuum of temperaments
So if 3\7 (three degrees of 7edo) is at one extreme and 2\5 (two degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators. Thus, between 3\7 and 2\5 you have (3+2)\(7+5) = 5\12, five degrees of 12edo:
If we carry this freshman-summing out a little further, new, larger edos pop up in our continuum.
(generators larger than this are proper)
L/s = 4
Temperaments above 5\12 on this chart are called "negative temperaments" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 8\19) and 1/4-comma (close to 13\31). As these tunings approach 3\7, the majors become flatter and the minors become sharper.
Temperaments below 5\12 on this chart are called "positive temperaments" and they include Pythagorean tuning itself (well approximated by 22\53) as well as superpyth temperaments such as 7\17 and 9\22. As these tunings approach 2\5, the majors become sharper and the minors become flatter. Around 9\22, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
5L 2s contains the pentatonic MOS 2L 3s and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either 7L 5s or 5L 7s.